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La notation hyper-E étendue (Extended Hyper-E notation; E# en abrégé) est une notation pour les grands nombres imaginée par Sbiis Saibian[1]. Il s'agit de la deuxième étape du système extensible-E.

## Définition

The extended hyper-E notation is based on the notation hyper-E and allows multiple hyperions to appear between each entry. The number of hyperions following entry an is represented by h(n). For the sake of this definition, #n is a shorthand for n successive hyperion marks. For example, a full expression would be written E(b)a1h(1)a2h(2)...h(n-1)anh(n). Saibian uses @ to indicate the rest of the expression.

• Rule 1. If there are no hyperions:
$$E(b)x = b^x$$.
• Rule 2. If the last entry is 1:
$$E(b) @ \#^{h(n-1)}{a_n}\#^{h(n)}1 = E(b) @ \#^{h(n-1)}{a_n}$$.
• Rule 3. If $$h(n-1)>1$$:
$$E(b) @ \#^{h(n-2)}{a_{n-1}}\#^{h(n-1)}{a_n} = E(b) @ \#^{h(n-2)}{a_{n-1}}\#^{h(n-1)-1}{a_{n-1}}\#^{h(n-1)}{a_n-1}$$.
• Rule 4. Otherwise:
$$E(b) @ \#^{h(n-2)}{a_{n-1}}\#{a_n} = E(b) @ \#^{h(n-2)}(E(b) @ \#^{h(n-2)}{a_{n-1}}\#{a_n-1})$$ (note $$\#^1$$ = $$\#$$).

We can also rewrite it in plain English:

1. If there is only one argument x, the value of the expression is bx.
2. If the last entry is 1, it may be removed.
3. Let h be the length of the last set of hyperion marks. If h > 1:
1. Remove the last entry of the expression and call it r.
2. Again remove the last entry of the expression; this time call it z.
3. Repeat "z" r times with h - 1 hyperion marks in between each repetition. Append this to the end of the expression. (Restore a removed hyperion mark sequence to glue the two expressions together.)
4. If the last set of hyperion marks is of length one:
1. Evaluate the original expression, but with the last entry decreased by 1. Call this value z.
2. Remove the last two entries of the expression.
3. Add z as an entry to the end of the expression. (Again, restore a removed hyperion mark sequence to glue the two expressions together.)

## Approximation

We evaluate the recursion level of this notation with the hiérarchie de croissance rapide (HCR) with the Wainer hierarchy.

As written in the notation hyper-E, we have relationship

Comparing the recusrion level, adding a # corresponds to adding an uparrow, which is in HCR adding 1. adding 2 #s corresponds to adding 2 in HCR, and adding n #s corresponds to adding n in HCR.

Two hyperion marks (deutero-hyperions), ##n, is expanded as n times repetitions of #, and therefore equivalent to n or ω in HCR. Repetitions of n ##s corresponds to ω×n.

Three hyperion marks (trito-hyperions), ###n, is expanded as n times repetitions of ##, and therefore equivalen to ω×n or ω2 in HCR. Repetitions of n ###s corresponds to ω2×n.

Therefore, roughly speaking, n+1 hyperion marks ##...## corresponds to ωn in HCR.

## Examples and googolisms

Some googolisms with this notation is shown for showing how the calculation proceeds.

• E100##100 = E100#100#100#...#100#100#100 with 100 repetitions of 100 = gugold
By using the rough estimation shown above, it is approximated as fω(100).
• E100##100##100 = E100##100#100#...#100#100 with 100 repetitions of 100 = gugolthra
We ignore the first ## until the second one has been expanded and all the 100s have been solved. By using the rough estimation shown above, it is approximated as fω×2(100).
• E100###100 = E100##100##...##100##100 with 100 repetitions of 100 = throogol
Three hyperion marks (trito-hyperions) constitute a repetition of two hyperion marks. Remember, the double marks are solved from right to left. It is approximated as fω2(100).
• E100####100 = E100###100###...###100###100 with 100 repetitions of 100 = teroogol
Quadruple hyperions decompose into triples. It is approximated as fω3(100).
• Godgahlah = E100#####...#####100 with 100 hyperion marks or E100#100100
Sets of 100 hyperion marks decompose into 99s, 99s decompose into 98s, etc. Also note that the superscript 100 means that there are 100 #'s, and should not be confused with E100#(100100). It is approximated as fω99(100) ≈ fωω(99).